There are a few minor practical problems with jet packs, or more properly, rocket packs as these devices are rockets, not turbojets, and it matters. It matters because there are some pretty fundamental limits on how much fuel/reaction mass a soldier can carry, especially when they have to carry other stuff like body armor, weapons, helmets, ammunition, food, shelter. Let's imagine that a stripped down soldier carrying little but armor, ammo and a rifle has a mass of 100 kilograms, 90 kilograms of which are the actual soldier. The rocket pack described above had a mass (full) of 57 kilograms and required the pilot to wear heat-protective clothing -- let's call it 60 kilograms. So wearing it and suitable armor and carrying weapons would be a roughly 70 kilogram burden on a 90 kilogram soldier, sort of like wearing a liquid nitrogen and hydrogen-peroxide-filled teen-ager on your shoulders as you wade into battle.
Sadly, this model would not work at all for the current rocket pack designs -- they provide less than 1500 N of thrust, and our soldier now has a weight of 1600 N. He would burn half of his fuel (give or take) waiting for the fuel levels in the tank to drop to where he could take off at all. The troop of rocket-equipped soldiers would all have to be "feather merchants" -- mass 70 kg or less -- and be armed with plastic squirt guns to get off of the ground at all.
Even with modern improvements, nobody has been able to increase flight time beyond around 30 seconds. The practical range in 30 seconds is perhaps 200 to 300 meters, at a height of ten meters -- a height great enough that it is already dangerous to fatal if one falls from it wearing an explosive, superheated massive outfit on your back. One cannot expect to increase their range or flight time because rockets eject mass backwards at high speed in order to provide thrust forward. The backward speed of the reaction mass is determined and limited by thermodynamics and chemistry and the need not to cook the soldier to extra crispy in a 30 second flight. There isn't that much variation in what's available to use for thrust in this context -- one could probably improve on the 740 C exhaust temperature, but only at the expense of adding a lot more shielding (and weight) and much more protective clothing.
The more interesting possibility is to build an actual jet pack -- jets of course use air for thrust mass and use fuel just to heat and compress the air, so they potentially have a much greater range. Small jet engines are mostly hobbyist stuff at the moment, but can produce order of a kilonewton of force at a mass cost of maybe 20 kilograms for the engine itself. One would need two, still further efficiency improvements, serious hearing protection, better shielding in the clothing (jet exhaust is still hotter than the "rocket" exhaust of hydrogen peroxide catalysis to water and oxygen). There is even military technology associated with cruise missiles that could be adapted.
We could learn another lesson from cruise missiles as well. Wings help. Wingsuits, for example, increase the glide ratio of skydiver to six. Hang gliders can achieve 17 to 20. Equipping a small hang glider with a small jet engine (one engineered to run without overheating for indefinite periods of time, unlike many of the powered hang glider engines currently available that tend to be based on two stroke chainsaw motors) could conceivably result in a wearable harness with a comparatively small wingspan in which a fully equipped soldier would have a range of tens of kilometers in tens of minutes at heights ranging from 10s of meters to a thousand meters or so. After powering up and attaining height, the engines could be shut off and the gliders could passively and silently descend from a height of a kilometer to a target 10 kilometers away.
The wings would have to be designed to be "compressible" to a comparatively small pack and quickly and easily erected into a structurally stable functional form, and would probably mass 50 to 70 kg including fuel, but this just puts them into the exact same range as rocket packs -- barely light enough to be useable by a soldier small enough to fly and large enough to lift it.
The good thing about this is that hang gliders are cheap, established technology, and small jet engines are straightforward to design and build, using readily available kerosene as fuel. The bad thing is that powered or not, a troop of soldiers in hang gliders would be a skeet shoot. As would soldiers with rocket packs.
There is a reason the military has focused on armed and armored helicopters to accomplish exactly the same purpose as rocket pack or jet powered wing, and has concentrated an entire branch of the military on the far more interesting problem of delivering death and destruction on a ground based enemy from the air or moving soldiers and material longer distances. It is a lot more cost effective, requires a lot fewer trained flight personnel, is vastly safer, provides enormously greater protection for the troops en route, has active offensive capabilities while in the air, and doesn't have to travel "with" the soldiers in the field to be of use as it is capable of going to them wherever they are.
So whether or not one can come up with imaginative rocket pack designs or more practical jet pack or jet wing designs, I'm pretty sure the military abandoned the idea because even if they worked perfectly at a nearly science fiction level the idea is dumb. Between the limitations imposed by physics and chemistry on any of the various design families and the fact that troops travelling slowly through the air using expensive, massive equipment that has to be kept with them at the front lines to be of any use might as well have big targets painted on them all the time that they are in the air, just dumb.
One could conceive of keeping a single unit in the army for "special ops" kinds of missions where they might be equipped with such things, and I wouldn't be at all surprised if they don't already have one. But equipping any significant fraction of front-line troops in a modern battlefield with such devices? Sheer madness.